MINIMAX TESTS FOR CONVEX CONES

LUTZ DÜMBGEN

Institut für Angewandte Mathematik, Universität Heidelberg, Im Neuenheimer Feld 294, D-69120 Heidelberg, Germany

(Received November 12, 1993; revised May 30, 1994)

Abstract. Let (P_{θ} : θ ∈ R^{p}) be a simple shift family of distributions on R^{p}, and let K ⊂ R^{p} be a convex cone. Within the class of nonrandomized tests of K versus R^{p} \ K , whose acceptance region A satisfies A = A + K , a test with minimal bias is constructed. This minimax test is compared to a likelihood ratio type test, which is optimal with respect to a different criterion. The minimax test is mimicked in the context of linear regression and one-sided tests for covariance matrices.

Key words and phrases: Bias, convex cone, covariance matrix, duality, linear regression, minimax test, union-intersection principle.